Most people here have heard about the Grandfather Paradox. Most people here have also read at least one time-travel story in which people either take out or attempt to take out Hitler. But if we have one, can we possibly do the other?
The Grandfather Paradox is as old as the concept of time travel. Here's the basics of it — if you were able to travel back in time, you'd be able to kill your grandfather before your father was ever born, which would lead to you never being born. But, of course, you never being born means that you couldn't possible have gone back in time and killed your grandfather in the first place. This means your grandfather lives, and so do you, and so you can go back in time with murderous intent — except you can't. Oh, time travel is such a puzzling thing!
The inventor of the paradox picked "grandfather," but it's understood that the paradox extends to anything that might negate one's own existence. Given the amount of different factors that we now know go into conception, this could be a lot of different things. If you drive past your grandparents house and startle them out of a romantic moment, half the DNA needed to create your father could simply, ah, die on the vine, so to speak. If your presence in Paris in the late 1800s happens to turn your great-great-grandfather's eyes towards the thighs of a can-can dancer with whom he would elope instead of marrying your great-great-grandmother, you would be paradoxing yourself. Yes, little packets of paradox could be anywhere.
Naturally, people question how exactly the paradox would work itself out. Those with a dramatic turn of mind might think of, say, a series of strange and startling coincidences steering you away from anything that might negate your existence. I prefer thinking of time travel as playing an online game with a very bad connection (or a lot of bugs). Make the wrong step, some step that would cause you to kill yourself off, and you flicker back to the future again. This both allows for time travel and upholds the paradox. It also explains why we aren't beset with time travelers, either now or at any of the historical events we find important. The farther you go back in history, and the more important the events you attend, the more people you are likely to influence. And the more people you influence, the more likely you are to nudge yourself out of existence through some silly chain of events that you can't possibly fully understand, and therefore can't avoid. Perhaps people, far into the future, have tried to come back and look around at our time but the connectivity problems don't make the trip worth it.
Which brings us from a practical conundrum to an ethical one. Another paradox most people are familiar with is whether it would be morally right for a present-day person to go back and kill Hitler before he had done anything worth being executed for. Would we be justified in killing an innocent person if we knew that person was going to be guilty? Especially if killing them early makes them die innocent, negating the justice of their execution? This ethical paradox has been debated by plenty of time-traveling scifi shows, but, if we take the Grandfather Paradox to heart, it's a practical problem as well. Who on Earth would we get to go? There's no denying that World War II changed the world so entirely that most people now would be disqualified from even attempting to kill Hitler.
Can anyone reading say that their birth wasn't influenced by the social, economic, and political changes brought on by World War II? And, if anything, the problem is going to get worse over time. Right now, it looks like that the only people who could stop World War II by killing Hitler are those who were born before World War II started. Although that would make for an amazing movie, it means we've only got a certain amount of time left to get time travel going before the most famous ethical time travel quandary to fall victim to the most famous practical one.
Of course, even if we ironed that out, there's a dark side to the Grandfather Paradox. You may not be able to negate your own existence, but anyone else (except your siblings or direct descendants) could. They could even by accident. Killing Hitler in this way would probably wipe out everyone in the world, and replace them with a completely different set of people. These people might be understandably freaked out by the idea that anyone around them could accidentally write them out of existence by waving hello to their mother at a crucial moment. How would a society like this protect itself?
The only way would be to make each new person's life directly influenced by the greatest number of existing people. The best way of everyone staying alive is to work out a society where everyone has as much influence over the sex life of everyone else as possible. So we'd not only take out a good chunk of the modern population of the world, we'd create a society with procreation by popular vote. Time travel is a funny thing.